Google Analytics is a free web analytics solution provided by Google, featuring several client-side APIs, as well as REST APIs for Data Export and for Management.
API's that are part of Google Analytics
Tracking Code API
This is a JavaScript Library that can be used by website developers/webmasters to track site activity sending user click path to the tool data pool.
http://code.google.com/apis/analytics/docs/tracking/home.html
Management API
REST API. Can be used to retrieve all management information on Google Analytics: Users, Profiles, Goals, Filters, etc.
http://code.google.com/apis/analytics/docs/mgmt/home.html
Data Export API
REST API. Can be used to retrieve all data that is present on the Google Analytics Interface. It's a read-only API.
http://code.google.com/apis/analytics/docs/gdata/home.html
Tools
Google provide some playground-like tools to let you play and learn the APIs